Ergodic averages and the large intersection property along IP sets

TL;DR

This paper investigates the convergence properties of multiple ergodic averages along IP sets, providing criteria for convergence, divergence, and the influence of nilsystems, with implications for large intersection properties.

Contribution

It introduces new criteria for mean and pointwise convergence of ergodic averages along IP sets and links their behavior to nilsystems and rational obstructions.

Findings

Criteria for convergence and divergence of averages

Convergence results for nilsystem-controlled sequences

Existence of large intersections along IP sets

Abstract

We study multiple ergodic averages along IP sets, meaning we restrict iterates in the averages to all finite sums of some infinite sequence of natural numbers. We give criteria for convergence and divergence in mean of these multiple averages and derive sufficient conditions for convergence to the projection onto the space of invariant functions. For a class of sequences that, roughly speaking, only have rational obstructions to such a limit, we show that the behavior is controlled by nilsystems. We also consider pointwise convergence, obtaining convergence and a formula for a set of functions on nilsystems that are dense in $L^2$. Finally, we show that certain correlations have optimally large intersections along an IP set